If the goal of training of a GAN is to have $P_g=P_{data}$, shouldn't this produce the exact same images?
Answers:  


No, generally GANs model the joint distribution of pixel space. The space is so entangled that it would be very difficult to model a singular pixels PDF.  
This question doesn't make sense given the first question's answer. You do NOT sample singular pixels, you sample the joint.  
$P_g$ is trying to model $P_{data}$, so that's your goal, so yes it will find features in the training set. Ideally, though it can extrapolate usage of the features to generate new images. But, yes depending on your setup you can end up with overfitting or mode-collapse where you really only generate a small subset of images that trick the discriminator. So I would say, no, the GAN doesn't know what its doing (since its just a minimax optimization of a loss you setup), so your goal is to tell it what to do in a way to achieve your best result.